I'm independently studying Boyd & Vandenberghe's Convex Optimization and came across the following statement, which I'm trying to prove.

The affine hull of a set $C$ is the smallest affine set that contains $C$.

where affine hull is defined as

$$\textrm{aff}\,C = \{\theta_1x_1 + \dots + \theta_kx_k \vert x_1, \dots, x_k \in C, \theta_1 + \dots + \theta_k = 1\}$$

and an affine set is defined as a set $S$ where for any $x_1, x_2 \in S$ and $\theta \in \mathbb{R}$, we have $\theta x_1 + (1-\theta)x_2 \in S$.

I was reading through the answer provided on this post which involves proving that the affine hull is an affine set by showing that the affine hull is "closed under affine combination". However, is this sufficient to proving that the affine hull is an affine set?

Specifically, does a set $D$ that is closed under affine combination imply that $D$ is an affine set? I understand that the implication "it can be shown that an affine set is closed under affine combination" is true, but I don't see how the converse is true.

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    $\begingroup$ What’s your definition of “affine set”? $\endgroup$ – David M. Feb 5 at 4:35
  • $\begingroup$ You might want to read about this in Rockafellar's Convex Analysis. I assume you do this to eventually learn about relative interiors. Rockafellar's book remains the standard reference on these topics. $\endgroup$ – max_zorn Feb 5 at 8:51
  • $\begingroup$ @DavidM., I updated the question with my definition of affine set. $\endgroup$ – Noah Stebbins Feb 5 at 13:22
  • $\begingroup$ Your provided definition seems to me to be the mathematical statement of “closed under affine combination” (i.e. you define an affine set to be a set closed under affine combination) $\endgroup$ – David M. Feb 5 at 13:28
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    $\begingroup$ @DavidM., wow can't believe I didn't see that. Thank you! $\endgroup$ – Noah Stebbins Feb 5 at 14:11

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